What is the interpretation of the Prophet's (peace be upon him) saying: "The greeting of the one walking is upon the one sitting," and is it obligatory to stand for a parent or elder if he is walking?

The younger greets the elder, the passerby greets the one sitting, the few greet the many, and the rider greets the walker. Scholars have mentioned the wisdom behind this: the greeting by the younger, the many, the passerby, and the rider is out of reverence and humility, to prevent arrogance, or to avert the expectation of evil, or due to lack of hardship.

As for standing up for an arriving person, scholars have differed regarding its permissibility. Some, like Imam Al-Nawawi, permitted it and even deemed it desirable for someone who possesses clear merit in knowledge, righteousness, nobility, or authority, provided this standing is for honor and respect, not for ostentation. Others disliked it except for a ruler, a scholar, a parent, a person of piety and devoutness, a respected person of the community, or an elderly person in Islam. Based on this, there is no objection to you standing up for your father or any other elder.